TET10 · quadratic tetrahedron

Physical formulationQUALIFIED_BOUNDED · nonlinear extensions remain narrower

Ten-node isoparametric solid with four corner nodes, six midside nodes, and thirty translational DOFs. Straight-sided cells use four-point Hammer stiffness integration; curved cells switch to a 64-point Duffy rule, while consistent mass uses 125 points.

01 Model and scope

Quadratic corner and edge functions interpolate both the curved geometry and displacement field. The chain rule maps reference derivatives to physical gradients through J⁻ᵀ; stiffness then integrates the resulting spatially varying B over the element.

Route assumptions and calculation checks

The geometry and displacement use the same quadratic basis; the six edge-node positions control whether the geometry is straight or curved.
The implementation selects four-point Hammer stiffness integration for straight-sided cells and 64-point Duffy integration for curved cells; consistent mass uses a separate 125-point rule.
Jacobian sign is checked at a closed sample lattice and at integration points; this is a strong diagnostic, not an analytic proof of positivity everywhere.

Useful foundations and prerequisites

Interpolation, mapping and assembly Strain, stress and rotation measures

02 Essential equations and symbols

N_i=L_i(2L_i-1),\quad i=1,\ldots,4

Definitions and conventions

Lᵢ: Barycentric coordinate at corner i.
Nᵢ, Nᵢⱼ: Quadratic corner and edge shape functions.
$J$: Jacobian of the isoparametric geometry map.
w_g, det(J_g): Quadrature weight and local volume scale at point g.

TET10 topology and integration pathThe midside coordinates participate in the isoparametric geometry map as well as the displacement field. The right-hand summary separates the straight-cell stiffness rule, curved-cell rule, and mass rule.

6 detailed steps

03 Step by step derivation

Follow the assumptions, equations and discrete operations in order. The full explanation stays visible.

At each integration point, J = ∂x/∂ξ maps parent coordinates to physical space. The chain rule gives ∇ₓN = J⁻ᵀ∇ξN, then B is assembled from these physical gradients. The local energy is integrated as Kₑ = Σg w_g det(J_g) B_gᵀ D B_g. Curvature makes J and B vary within the cell, so a valid Jacobian and adequate quadrature are essential.

1 · Ten nodes and the quadratic tetrahedral basis

TET10 adds one node at the midpoint of each of the six edges to the four TET4 corners. The documented local edge order is (1,2), (2,3), (3,1), (1,4), (2,4), (3,4); this ordering is part of the mesh-connectivity contract. Each node carries three translations, so the element vector has 30 entries.

N_i=L_i(2L_i-1),\quad i=1,\ldots,4

Permuting midside nodes changes both geometry and displacement interpolation. It is not just a display-order issue.

2 · Isoparametric geometry and displacement

The same quadratic shape functions map the reference tetrahedron into the physical cell and interpolate displacement. This lets the midside coordinates represent curvature. If all midside points lie exactly on straight edges, the geometry is straight even though the displacement basis remains quadratic.

\mathbf x(\boldsymbol\xi)=\sum_{a=1}^{10}N_a(\boldsymbol\xi)\mathbf x_a

For an affine geometry map the quadratic basis reproduces quadratic displacement fields. Curved geometry introduces a varying Jacobian and changes the physical gradients at every integration point.

3 · Physical gradients, strain, and the stiffness integral

At each quadrature point the implementation computes the Jacobian, rejects a non-positive determinant, maps reference derivatives into physical derivatives, then builds the same engineering-shear B convention as TET4. Unlike TET4, B and det(J) generally vary through the cell, so stiffness is a weighted sum rather than a single volume product.

\nabla_{\!x}N_a=\mathbf J^{-T}\nabla_{\!\xi}N_a,\qquad \boldsymbol\varepsilon_g=\mathbf B_g\mathbf q_e

Straight-sided cells use the four-point Hammer degree-two rule. Curved cells switch to a positive 64-point Duffy rule of order four; the 125-point Duffy rule belongs to consistent mass. A closed 35-point barycentric Jacobian sample detects bad geometry, but sampling is not an analytic proof that det(J)>0 everywhere.

4 · Consistent mass and quadratic face loads

The consistent mass is the integral of the interpolated velocity field, including off-diagonal coupling between the ten nodes. The implementation uses 5×5×5=125 Duffy/Gauss points for this volume integral. A quadratic triangular face has three corners and three midside nodes; a uniform traction can therefore assign zero force to corner nodes and nonzero equivalent force to midside nodes without losing resultant or moment.

\mathbf M_e=\int_{\Omega_e}\rho\,[\mathbf N^T\mathbf N\otimes\mathbf I_3]\,d\Omega

Positive density is required for the modal mass. The face rule and the geometry map are part of the load calculation, not just the solid stiffness kernel.

5 · What the verification results say

Element checks cover partition of unity, derivatives, six rigid modes, stiffness symmetry and energy, mass positivity, aligned and oblique affine patches, quadratic-field recovery, and Gmsh reorientation. Structural cases then probe bending, curved-cylinder Lamé response, modal frequencies, loads, and same-mesh external correlation. In the cited four-level bending campaign, final compliance error was 45.75% for TET4 and 1.179% for TET10; that is one benchmark sequence, not a universal accuracy ratio.

\sum_aN_a\mathbf x_a=\mathbf x\quad\text{for affine geometry}

Evidence is observable-specific. For example, the curved-cell quadrature error compares against an order-eight Duffy reference; it does not establish every mesh, constitutive law, or nonlinear solve.

6 · Limits and route boundaries

Quadratic interpolation improves representation but does not remove the need for mesh quality, integration checks, or refinement. Curved midside placement must be geometrically intentional. Stress recovery between integration points is post-processing. The documented stateful nonlinear route currently stores states at the Hammer points; the curved-geometry enriched stiffness rule is outside that stateful quadrature path. Read the exact registry row before transferring evidence to nonlinear, modal, or contact combinations.

04 Algorithm and implementation

Source excerpt · v0.2.11

At each quadrature point, reference derivatives form the Jacobian. Its inverse maps gradients into physical coordinates; non-positive Jacobians are rejected.

dnr = cls.shape_derivatives_reference(barycentric)
jacobian = dnr.T @ coords
det_j = float(np.linalg.det(jacobian))
if det_j <= 1.0e-14:
    raise ValueError(f"Invalid TET10 jacobian {det_j:.6e}.")
gradients = dnr @ np.linalg.inv(jacobian).T
return strain_displacement_from_gradients(gradients), det_j

Read the complete versioned implementation →

Implementation